Solve the following equation : `(x^(2)-3.5 x + 1.5)/(x^(2)-x-6)=0`

Answer

To solve the equation

\[ \frac{x^2 - 3.5x + 1.5}{x^2 - x - 6} = 0, \]

we follow these steps:

### Step 1: Set the Numerator Equal to Zero
For a fraction to equal zero, the numerator must be zero (as long as the denominator is not zero). Therefore, we set the numerator equal to zero:

\[ x^2 - 3.5x + 1.5 = 0. \]

### Step 2: Solve the Quadratic Equation
To solve the quadratic equation \(x^2 - 3.5x + 1.5 = 0\), we can use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \]

where \(a = 1\), \(b = -3.5\), and \(c = 1.5\).

Calculating the discriminant:

\[ b^2 - 4ac = (-3.5)^2 - 4 \cdot 1 \cdot 1.5 = 12.25 - 6 = 6.25. \]

Now substituting back into the quadratic formula:

\[ x = \frac{3.5 \pm \sqrt{6.25}}{2 \cdot 1} = \frac{3.5 \pm 2.5}{2}. \]

Calculating the two possible values for \(x\):

1. \(x = \frac{3.5 + 2.5}{2} = \frac{6}{2} = 3\),
2. \(x = \frac{3.5 - 2.5}{2} = \frac{1}{2} = 0.5\).

### Step 3: Check the Denominator
Next, we need to ensure that these solutions do not make the denominator zero. The denominator is:

\[ x^2 - x - 6 = 0. \]

Factoring the denominator, we get:

\[ (x - 3)(x + 2) = 0. \]

This gives us the solutions \(x = 3\) and \(x = -2\).

### Step 4: Exclude Invalid Solutions
Since \(x = 3\) makes the denominator zero, it is not a valid solution. Thus, we only have:

\[ x = 0.5. \]

### Final Answer
The solution to the equation is:

\[ \boxed{0.5}. \]

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